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\author{王立庆（2021级-2022级数学与应用数学1班）}
\title{实变函数教学大纲（学生使用）}
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%\date{2023 年 9 月 1 日}

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\section*{时间地点}
\begin{enumerate}
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\item 上课时间地点：周一下午12:55 - 3:25（5-7节），六教220.
\item 答疑时间地点：周一晚上20:30 - 21:30, 周二下午13:00 - 17:00, 一教210. 
\end{enumerate}

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\section*{使用教材}
\begin{enumerate}\itemsep0em 
\item 程其襄、张奠宙、胡善文、薛以锋，实变函数与泛函分析基础，高等教育出版社，2019年6月第四版。
\end{enumerate}

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\section*{参考文献}
\begin{enumerate}\itemsep0em 
\item  胡善文、薛以锋，实变函数与泛函分析基础（第四版）学习指导与习题解答，高等教育出版社，2021年4月第一版。
\item 夏道行、吴卓人、严绍宗、舒五昌，实变函数论与泛函分析（上册），高等教育出版社，2010年1月第二版。
\item A. N. Kolmogorov, S. V. Fomin, Introductory Real Analysis, Translated and Edited by R. A. Silverman, Dover Publications, New York, 2015. 

\end{enumerate}

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\section*{课程成绩}
\begin{enumerate}
\item  平时成绩 100\%.%包括课堂考勤、课外作业、期中考试、阶段测验。
\begin{enumerate}\itemsep0em 
\item[1.1.] 课堂考勤10次，共20分。
\item[1.2.] 课外作业10次，共30分。
\item[1.3.] 期中考试1次，共20分。
\item[1.4.] 期末考试1次，共30分。
\end{enumerate}

%\item  期末成绩 60\%, 计算题和应用题。

\end{enumerate}

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\section*{主要内容}
\begin{enumerate}\itemsep0em 
\item  第一章：集合。集合运算法则，德摩根公式，集合序列的上下极限，伯恩斯坦定理，并集、直积与幂集的基数。可数基数与连续基数的关系。
\item  第二章：点集。开集与闭集的性质，直线上的开集、闭集与完备集的构造，$\mathbb{R}^n$ 中的开集的构造，紧集等价于有界闭集，康托尔三分集的性质。
\item  第三章：测度论。外测度的性质，勒贝格测度空间，博雷尔测度空间，勒贝格可测集的性质，勒贝格测度的性质，勒贝格可测集与博雷尔可测集的关系。
\item  第四章：可测函数。可测函数在四则运算、上下确界、上下极限运算下封闭，可测函数是简单函数列的极限，叶戈罗夫定理，卢津定理，里斯定理，勒贝格定理。
\item  第五章：积分论。勒贝格积分是线性泛函，勒贝格积分的绝对连续性，莱维定理，逐项积分定理，法图引理，勒贝格控制收敛定理，逐项收敛定理，黎曼可积的充要条件，富比尼定理。
\item  **第六章：微分与不定积分。单调函数几乎处处可导，若尔当分解定理，绝对连续函数是不定积分，绝对连续函数导数为零的必要条件，黎曼-斯蒂尔切斯积分存在的充分条件。

\end{enumerate}

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%\section*{重要定理}
%\begin{table}[ht!]\centering
%\begin{tabular}{|p{1cm}|p{12cm}|}  \hline 
% 章节 & 冠名定理 \\ \hline 
% 1.3 & 伯恩斯坦定理 \\ \hline
%% 2.2 & 波尔查诺-魏尔斯特拉斯定理 \\ \hline
%% 2.3 & 海涅-博雷尔有限覆盖定理 \\ \hline
% 2.4 & 开集构造定理  \\ \hline
% 3.2 & 卡拉泰奥多里条件 \\ \hline
% 4.2 & 叶戈罗夫定理 \\ \hline
% 4.3 & 卢津定理 \\ \hline
% 4.4 & 里斯定理 \\ \hline
% 5.3 & 莱维定理 \\ \hline
% 5.3 & 法图引理 \\ \hline
% 5.4 & 勒贝格控制收敛定理 \\ \hline
% 5.6 & 富比尼定理 \\ \hline
% 6.1 & 维塔利覆盖定理 \\ \hline
%\end{tabular}
%\end{table}

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\newpage
\section*{授课计划}

\begin{longtable}%[ht!]%\centering
%\begin{tabular}
{|p{1cm}|p{1cm}|p{8cm}|p{4cm}|}  \hline 
周 & 章节 & 内容 &习题 			\\ \hline \hline
1  & 1 & 集合 & 1, 4, 7, 10, 14, 17   \\  \hline
    & 1.1 & 集合的表示 &    \\  \hline
    & 1.2 & 集合的运算  &    \\  \hline
2  & 1.3 & 对等与基数、伯恩斯坦定理 &    \\  \hline
  & 1.4 & 可数集合 &    \\  \hline
    & 1.5 & 不可数集合 &    \\  \hline\hline
3  & 2 & 点集 & 1, 3, 5, 11, 12    \\  \hline
   & 2.1 & 度量空间、n维欧氏空间 &    \\  \hline
   & 2.2 & 聚点、内点、界点 &    \\  \hline
   & 2.3 & 开集、闭集、完备集 &    \\  \hline
   & 2.4 & 直线上的开集、闭集、完备集的构造 &    \\  \hline
   & 2.5 & 康托尔三分集 &    \\  \hline\hline
4  & 3 & 测度论 & 2, 4, 11, 13, 16   \\  \hline
    & 3.1 & 外测度 &    \\  \hline
    & 3.2 & 可测集、卡拉泰奥多里条件 &    \\  \hline
5  & 3.3 & 可测集类 &    \\  \hline
   & 3.4 & 不可测集 &    \\  \hline\hline
6 & 4 & 可测函数 & 1, 3, 6, 9, 13, 16   \\  \hline
   & 4.1 & 可测函数及其性质 &    \\  \hline
7 & 4.2 & 叶戈罗夫定理 &    \\  \hline
   & 4.3 & 可测函数的构造、卢津定理 &    \\  \hline
8  & 4.4 & 依测度收敛、里斯定理、勒贝格定理 &    \\  \hline\hline
9  & & {\color{red}期中考试} & \\  \hline\hline
10 & 5 & 积分论 & 1, 7, 12, 13, 14, 24, 25    \\  \hline
     & 5.1 & 黎曼积分的局限性、勒贝格积分简介 &    \\  \hline
     & 5.2 & 非负简单函数的勒贝格积分 &    \\  \hline
11 & 5.3 & 非负可测函数的勒贝格积分、莱维定理、法图引理 &    \\  \hline
12  & 5.4 & 一般可测函数的勒贝格积分、勒贝格控制收敛定理 &    \\  \hline
13 & 5.5 & 黎曼积分和勒贝格积分 &    \\  \hline
  & 5.6 & 勒贝格积分的几何意义、富比尼定理 &    \\  \hline\hline
14  & 6* & 微分与不定积分 &    \\  \hline
  & 6.1* & 维塔利定理 &    \\  \hline
  & 6.2* & 单调函数的可微性 &    \\  \hline
  & 6.3* & 有界变差函数、若尔当分解定理 &    \\  \hline
  & 6.4* & 不定积分 &    \\  \hline
  & 6.5* & 斯蒂尔切斯积分 &    \\  \hline
  & 6.6* & L-S测度与积分 &    \\  \hline\hline
15 & & {\color{red}期末考试} & \\  \hline
%\end{tabular}
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